The clubs pledged to work closely with police investigating the incident which came after Arouna Kone struck in the second minute to put the Toffees ahead at Brunton Park
The goal set Everton on the way to a 3-0 victory in which Aaron Lennon and Ross Barkley also scored
A public address system message soon after Kone's goal announced any racist abuse would not be tolerated.
Martinez said it was his understanding some "racial remarks" had come from the Warwick Road End stand as Kone and Steven Pienaar celebrated the opening goal, and that both were "hurt" by what they heard.
A statement from Cumbria Police issued after the game said: "Cumbria Constabulary are investigating allegations of racist behaviour...directed towards Everton players following the first goal.
"Officers are making enquiries and the force is asking for those with information to get in touch."
Carlisle boss Curle said in his post-match press conference: "It is not what we want inside Brunton Park, inside football, or inside society.
"The club will make all efforts
I know there was a steward in the terraces and CCTV as well
We will do everything we can to identify who it was and make sure the relevant penalties and punishments will be adhered to
"I didn't hear it myself, but what I would say to any supporters who were next to people and heard people - please step forward and identify them, because it is not accepted."
Everton manager Martinez said: "We will help the police, Carlisle and everyone to make sure that it doesn't happen again."
Asked how Kone and Pienaar were feeling, he said: "They are okay
"I think they were sensitive and hurt at that moment - probably they were fearful about the game developing into a difficult afternoon.
"But it was a one-off incident.
"As you know, there is no room for that sort of behaviour
But I must stress it was a very small minority and I want to believe that it was just an emotional reaction to the disappointment of conceding a goal."
Martinez said the incident was well handled by referee Lee Mason, who had been approached by one of the players and then relayed that to the managers, leading to the announcement being made.
Martinez was also keen to praise his team as Everton bounced back from the disappointment of their midweek Capital One Cup semi-final exit at Manchester City.
A Toffees side showing six changes from that game were 2-0 up inside the opening 15 minutes here through Kone and Lennon's close-range efforts, and sealed the victory in the 65th through Barkley's deflected strike from outside the box.
Martinez said: "I thought we started with a very professional approach and a really strong focus.
"Scoring the two goals early allowed us to manage the game well, and we did that with an incredible maturity which is always pleasing."
Curle, whose Sky Bet League Two side were the lowest-ranked outfit left in the competition, said: "Giving a Premier League side a two-goal start was not the ideal scenario, but credit to the players - they didn't melt or fold.
"And you have to say, Everton were fantastic."
A bumper crowd turned out for what was only Carlisle's second game back at their own stadium since last month's devastating floods, which had left the pitch at that time completely under water.
And Curle added: "A lot of credit has to go to a lot of people behind the scenes who worked to get this fixture on.
"To have 17,000 people inside Brunton Park was phenomenal.
"I think it was only 57 or 58 days ago that the place was decimated because of the disaster of the flood, so to be hosting football - and hosting Everton - here was excellent."
The Football Association on Sunday evening confirmed it was investigating the alleged racist abuse of Everton players.
A statement read: "We have received the referee's report regarding the alleged incident during today's game between Carlisle United and Everton
We have contacted both clubs this evening and will be conducting an immediate and thorough investigation.
"The FA strongly condemns any discriminatory behaviour and will continue to work with partners across the game to combat it whenever it arises."
Source : PA
Source: PA